Giả sử \(f(x)\) và \(g(x)\) là các hàm số bất kỳ liên tục trên \(\mathbb{R}\) và \(a,\,b,\,c\) là các số thực. Mệnh đề nào sau đây sai?
 | \(\displaystyle\int\limits_{a}^{b}f(x)\mathrm{\,d}x+\displaystyle\int\limits_{b}^{c}f(x)\mathrm{\,d}x+\displaystyle\int\limits_{c}^{a}f(x)\mathrm{\,d}x=0\) |
 | \(\displaystyle\int\limits_{a}^{b}cf(x)\mathrm{\,d}x=c\displaystyle\int\limits_{a}^{b}f(x)\mathrm{\,d}x\) |
 | \(\displaystyle\int\limits_{a}^{b}f(x)g(x)\mathrm{\,d}x=\displaystyle\int\limits_{a}^{b}f(x)\mathrm{\,d}x\cdot \displaystyle\int\limits_{a}^{b}g(x)\mathrm{\,d}x\) |
 | \(\displaystyle\int\limits_{a}^{b}\left( f(x)-g(x)\right) \mathrm{\,d}x+\displaystyle\int\limits_{a}^{b}g(x)\mathrm{\,d}x=\displaystyle\int\limits_{a}^{b}f(x)\mathrm{\,d}x\) |
Chọn phương án C.
\(\begin{align*}♥ &\displaystyle\int\limits_{a}^{b}f(x)\mathrm{\,d}x+\displaystyle\int\limits_{b}^{c}f(x)\mathrm{\,d}x+\displaystyle\int\limits_{c}^{a}f(x)\mathrm{\,d}x\\
=&\displaystyle\int\limits_{a}^{c}f(x)\mathrm{\,d}x+\displaystyle\int\limits_{c}^{a}f(x)\mathrm{\,d}x\\
=&\displaystyle\int\limits_{a}^{a}f(x)\mathrm{\,d}x=0.\end{align*}\)
\(♥ \displaystyle\int\limits_{a}^{b}cf(x)\mathrm{\,d}x=c\displaystyle\int\limits_{a}^{b}f(x)\mathrm{\,d}x\) (\(c\) là số thực).
\(\begin{align*}♥ &\displaystyle\int\limits_{a}^{b}\left( f(x)-g(x)\right) \mathrm{\,d}x+\displaystyle\int\limits_{a}^{b}g(x)\mathrm{\,d}x\\
=&\displaystyle\int\limits_{a}^{b}\left( f(x)-g(x)+g(x)\right)\mathrm{\,d}x\\
=&\displaystyle\int\limits_{a}^{b}f(x)\mathrm{\,d}x.\end{align*}\)